# Stefan–Boltzmann constant

(Redirected from Stefan's constant)

The Stefan–Boltzmann constant (also Stefan's constant), a physical constant denoted by the Greek letter σ, is the constant of proportionality in the Stefan–Boltzmann law: the total energy radiated per unit surface area of a black body in unit time is proportional to the fourth power of the thermodynamic temperature.

The value of the Stefan–Boltzmann constant is given in SI units by

σ = 5.670373(21)×10−8 W m−2 K−4.[1]

In cgs units the Stefan–Boltzmann constant is:

$\sigma \approx 5.6704 \times 10^{-5}\ \textrm{erg}\,\textrm{cm}^{-2}\,\textrm{s}^{-1}\,\textrm{K}^{-4}.$

In US customary units the Stefan–Boltzmann constant is:[2]

$\sigma = 0.1714 \times 10^{-8}\ \textrm{BTU}\,\textrm{hr}^{-1}\,\textrm{ft}^{-2}\,\textrm{R}^{-4}.$

The value of the Stefan–Boltzmann constant is derivable as well as experimentally determinable; see Stefan–Boltzmann law for details. It can be defined in terms of the Boltzmann constant as:

$\sigma = \frac{2\pi^5k_{\rm B}^4}{15h^3c^2} = \frac{\pi^2k_{\rm B}^4}{60\hbar^3c^2} = 5.670373(21) \, \cdot 10^{-8}\ \textrm{J}\,\textrm{m}^{-2}\,\textrm{s}^{-1}\,\textrm{K}^{-4}$

where:

The CODATA recommended value is calculated from the measured value of the gas constant:

$\sigma = \frac{2 \pi^5 R^4}{15 h^3 c^2 N_{\rm A}^4} = \frac{32 \pi^5 h R^4 R_{\infty}^4}{15 A_{\rm r}({\rm e})^4 M_{\rm u}^4 c^6 \alpha^8}$

where:

A related constant is the radiation constant (or radiation density constant) a which is given by:[3]

$a = \frac{4\sigma}{c} = 7.5657 \times 10^{-15} \textrm{erg}\,\textrm{cm}^{-3}\,\textrm{K}^{-4} = 7.5657 \times 10^{-16} \textrm{J}\,\textrm{m}^{-3}\,\textrm{K}^{-4}.$

A simple rule to remember the Stefan–Boltzmann constant is to think "5-6-7-8;" and try not to forget the negative sign before the final eight.

## References

1. ^ "CODATA Value: Stefan-Boltzmann constant". The NIST Reference on Constants, Units, and Uncertainty. US National Institute of Standards and Technology. June 2011. Retrieved 2011-06-23.
2. ^ Heat and Mass Transfer: a Practical Approach, 3rd Ed. Yunus A. Çengel, McGraw Hill, 2007
3. ^ Radiation constant from ScienceWorld